\newproblem{lay:7_4_23}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 7.4.23}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Given the Singular Value Decomposition theorem:\\
	
	\fbox{\parbox{0.9\linewidth}{
		Let $A\in\mathcal{M}_{m\times n}$ be a matrix with rank $r$. Then, there exists a matrix $\Sigma\in\mathcal{M}_{m\times n}$ whose diagonal entries
		are the first $r$ singular values of $A$ sorted in descending order ($\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_r > 0$) and there exist
		orthogonal matrices $U\in\mathcal{M}_{m\times m}$ and $V\in\mathcal{M}_{n\times n}$ such that
		\begin{center}
			$A=U\Sigma V^T$
		\end{center}
	}}
	\\
	Let $U=\begin{pmatrix}\mathbf{u}_1 & \mathbf{u}_2 & ... & \mathbf{u}_m\end{pmatrix}$ and $V=\begin{pmatrix}\mathbf{v}_1 & \mathbf{v}_2 & ... & \mathbf{v}_n\end{pmatrix}$.
	Show that 
	\begin{center}
		$A=\sigma_1\mathbf{u}_1\mathbf{v}_1^T+\sigma_2\mathbf{u}_2\mathbf{v}_2^T+...+\sigma_r\mathbf{u}_r\mathbf{v}_r^T$
	\end{center}
}{
   % Solution
	If we expand the SVD, we have
	\begin{center}
		$\begin{array}{rcl}
			A&=&U\Sigma V^T\\
			   &=&\begin{pmatrix}\mathbf{u}_1 & \mathbf{u}_2 & ... & \mathbf{u}_m\end{pmatrix}
				    \begin{pmatrix}\sigma_1 & 0 & ... & 0 & 0 & ... & 0 \\
						               0 & \sigma_2 & ... & 0 & 0 & ... & 0 \\
													 ...&... & ... & ... &... &... & ... \\
													 0 & 0 & ... & \sigma_r & 0 & ... & 0 \\
													 0 & 0 & ... & 0 & 0 & ... & 0 \\
													 ...&... & ... & ... &... &... & ... \\
													 0 & 0 & ... & 0 & 0 & ... & 0 \\
						\end{pmatrix}
						\begin{pmatrix}\mathbf{v}_1^T \\ \mathbf{v}_2^T \\ ... \\ \mathbf{v}_n^T \end{pmatrix}\\
			   &=&\begin{pmatrix}\sigma_1\mathbf{u}_1 & \sigma_2\mathbf{u}_2 & ... & \sigma_r\mathbf{u}_r & \mathbf{0} & ... & \mathbf{0}\end{pmatrix}
				    \begin{pmatrix}\mathbf{v}_1^T \\ \mathbf{v}_2^T \\ ... \\ \mathbf{v}_n^T \end{pmatrix}\\
			   &=&\sigma_1\mathbf{u}_1\mathbf{v}_1^T+\sigma_2\mathbf{u}_2\mathbf{v}_2^T+...+\sigma_r\mathbf{u}_r\mathbf{v}_r^T\\
		\end{array}$
	\end{center}
}
\useproblem{lay:7_4_23}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}

